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Actions of Z/2n on S3

Published online by Cambridge University Press:  05 March 2012

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Summary

ABSTRACT

This paper is devoted to classifying the actions of the cyclic group Z/2n on the 3-sphere S3. In particular, we show that if h ∈ Z/2n is a generator, then h is equivalent to a standard rotation of S3 if and only if h is a free action or has an almost tame fixed point set.

INTRODUCTION

The object of this paper is to classify all actions of the cyclic groups Z/2n on the 3-sphere S3. In section 2 we classify the non-free actions and in section 3 the free actions of Z/2n. In 1970, F. Waldhausen [19] has proven that every even periodic P.L. homeomorphism of S3 with 1-dimensional fixed point set is topologically equivalent to a standard rotation of S3. We shall extend Waldhausen's result by omitting the P.L. hypothesis and showing that every even periodic homeomorphism of S3 with non-empty fixed point set is topologically a standard rotation if and only if the set of fixed points is almost tame. In view of R. H. Bing's work [1] and [2], this is the strongest possible generalization of Waldhausen's result. This also settles the Smith conjecture [3] for homeomorphisms of period 2n whose fixed point sets are almost tame knots.

The problem of characterizing free cyclic actions on S3 remains largely unsolved. Thus far only free actions of Z/2 [6], Z/4 [11], and Z/8 [12] have been classified.

Type
Chapter
Information
Transformation Groups
Proceedings of the Conference in the University of Newcastle upon Tyne, August 1976
, pp. 147 - 153
Publisher: Cambridge University Press
Print publication year: 1977

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